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Interest Rate Modelling

25857 Interest Rate Modelling

UTS Business SchoolUniversity of Technology Sydney

Chapter 22. Modelling Interest Rate DynamicsMay 17, 2014

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Chapter 22. Modelling Interest Rate Dynamics

1 The Relationship between Interest Rates, Bond Prices and Forward Rates

2 Modelling the Spot Rate of Interest

3 Motivating the Feller (or Square Root) Process

4 Fubini’s Theorem

5 Modelling Forward RatesFrom Forward Rate to Bond Price DynamicsA Specific Example

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The Relationship between Interest Rates, Bond Prices and Forward Rates

The Relationship between Interest Rates, Bond Prices and

Forward Rates

In this section we shall clarify the relationship betweeninterest rates, bond prices, yield to maturity andforward rates.

Let P (t, T ) denote the price at time t of a pure discount bondpaying $1 at time T .

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Forward Rates

The yield to maturity �(t, T ) is the continuously compoundedrate of return causing the bond price to satisfy the maturitycondition

P (T, T ) = 1, (1)

that is, �(t, T ) satisfies (see Fig.1)

P (t, T )e�(t,T )(T−t) = 1. (2)

∼ The yield may also be expressed as

�(t, T ) = − lnP (t, T )

(T − t). (3)

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Forward Rates

time

Bond price

t T

P (t, T )

$1

b

b

�(t, T )

0

Figure 1: The yield to maturity �(t, T ); satisfies P (t, T )e�(T−t) = 15 / 77




Forward Rates

The instantaneous spot rate of interest, r(t), is the yield onthe currently maturing bond, i.e.,

r(t) = �(t, t). (4)

∼ By allowing t → T in (3)1

r(T ) = Pt(T, T ), (5)

where Pt(T, T ) denotes the partial derivative of P (t, T ) withrespect to its first argument (running time t), evaluated at thetime t = T .

1We need to apply L’Hopital’s rule since setting t = T in (3) yields themeaningless ratio 0/0. Thus by L’Hopital’s rule

limt→T

�(t, T ) = limt→T

−

∂∂t

lnP (t, T )

∂∂t

(T − t)= lim

t→T

−1P (t,T )

Pt(t, T )

−1= lim

t→T

Pt(t, T )

P (t, T )= Pt(T, T ).

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Forward Rates

The forward rate arises when we consider an investor whoholds a bond maturing at T1 and asking

∼ what return he or she would earn between T1 and T2(> T1), ifhe or she contracted now at time t.

∼ The required rate of return is the forward rate f(t, T1, T2)defined by

P (t, T1) = P (t, T2)ef(t,T1,T2)(T2−T1),

i.e.

f(t, T1, T2) =1

T2 − T1ln

[

P (t, T1)

P (t, T2)

]

. (6)

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Forward Rates

To see how f(t, T1, T2) represents the implicit rate ofinterest currently available (at time t) on riskless loansfrom T1 to T2 consider the set of transactions illustrated inFig.3.

∼ A set of transactions at time t involving zero net cashflow hasthe net effect of investing P (t, T2)/P (t, T1) dollars at time T1

to yield a dollar for sure at time T2.∼ The forward rate defined above is simply the continuously

compounded rate of interest earned on this investment.

Note that calculation of f(t, T1, T2) involves only bond pricesobservable at t.

Fig.2 displays a time line that is useful where thinking aboutthe relation between forward rates and bond prices.

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Forward Rates

b b bt T1 T2time

f(t, T1, T2)

P (t, T1)

P (t, T2)

Figure 2: The forward rate f(t, T1, T2) and bond prices P (t, T1), P (t, T2)

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Forward Rates ---t T1 T2Buy 1 bondmaturing T2Cash ow = �P (t; T2)Short sell P (t;T2)P (t;T1)bonds maturing T1Cash ow = +P (t; T2)Net Cash ow= 0 Cash ow = �P (t;T2)P (t;T1)(to over maturingbond) Cash ow= +1

1

Figure 3: The net zero transaction that relates the forward f(t, T1, T2)and bond prices P (t, T1) and P (t, T2)

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Forward Rates

The Heath-Jarrow-Morton model to be discussed in a laterchapter focuses on the instantaneous forward rate defined by2

f(t, T ) ≡ f(t, T, T ), (7)

∼ is the instantaneous rate of return the bond holder can earn byextending the investment an instant beyond T .

2There is a slight abuse of notation in that we use the symbol f to denoteboth the three variable function f(t, T1, T2) and the two variable functionf(t, T ).

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Forward Rates

Letting T2 → T1, in (6)3

f(t, T ) = −PT (t, T )

P (t, T ). (8)

3It is best to set T1 = T , T2 = T + x in (6) and calculate as follows:

f(t, T ) = limx→0

1

x[lnP (t, T )− lnP (t, T + x)] = lim

x→0−

∂∂x

lnP (t, T + x)∂∂x

x

= limx→0

−PT (t, T + x)

P (t, T + x)=

−PT (t, T )

P (t, T ).

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Forward Rates

It follows from (8) that the price of the pure discount bondmay be written in terms of the forward rate as

P (t, T ) = e−∫ Tt

f(t,s)ds. (9)

In a world of certainty all securities, in equilibrium, mustearn the same instantaneous rate of return so as to excludethe possibility of riskless arbitrage opportunities.

∼ The application of this equilibrium condition to discount bondsimplies

Pt(t, T )

P (t, T )= r(t), (10)

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Forward Rates

from which4 we obtain the relationship between pure discountbond prices and the instantaneous spot rate in a world ofcertainty,

P (t, T ) = e−∫ T

tr(s)ds. (11)

4From (10)d

dslnP (s, T ) = r(s).

Integrating (t, T ) we obtain

ln[P (s, T )]Tt = lnP (T, T )−lnP (t, T ) = ln 1−lnP (t, T ) = − lnP (t, T ) =

∫ T

t

r(s)ds.

Hence (11) follows.14 / 77




Forward Rates

A comparison of (11) and (2) shows the relationship betweenthe yield to maturity and the spot rate in a world of certainty,

�(t, T ) =1

(T − t)

∫ T

tr(s)ds. (12)

Next, substituting (11) into (8) reveals the relationshipbetween the spot rate and the forward rate in a world ofcertainty viz.

f(t, T ) = r(T ), for all T ≥ t. (13)

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Forward Rates

This last eqn. is a degenerate version of the expectationshypothesis,

∼ i.e., the expected instantaneous spot rate for time T is equal tothe instantaneous forward rate for time T , calculated at time t.

In subsequent chapters we shall see how the relationships(11), (12) and (13) generalise quite naturally in a world ofuncertainty to be the corresponding relationships undersuitable probability measures.

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Forward Rates

Another important set of interest rates are LIBOR rates.

Akin to the forward rates f(t, T1, T2):

∼ Also rates that one can contract at time t for borrowing over afixed period in the future.

∼ However the time difference between current time and thatfixed period in the future remains constant.

∼ With f(t, T1, T2) the time difference between t and T1

decreases as time evolves.∼ In fact it would make no sense to consider t beyond T1.

We shall discuss LIBOR rates in Chapter 26 when we discussthe Brace-Gatarek-Musiela (BGM) LIBOR market model.

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Modelling the Spot Rate of Interest


Consider models for the dynamics of the spot rate of interest.

∼ A number of proposed models are of the general form

dr = �(r, t)dt+ �(r, t)dW. (14)

Typically the drift term is of the form

�(r, t) = �(r − r), (15)

∼ where r is the long run level of the spot rate of interest. Thisform implies mean reverting behaviour of the spot rate ofinterest which is confirmed in the empirical studies discussedbelow.

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It is less clear what form the volatility function should take.

∼ Forms usually used in empirical studies and in development ofterm structure and interest rate derivative models to bediscussed later are of the general form

�(r, t) = �r , ≥ 0. (16)

Interest rate process becomes

dr = �(r − r)dt+ �r dW. (17)

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0 0.1 0.2 0.3 0.4 0.5

0

0.02

0.04

0.06

0.08

Sample paths of r(t) with γ = 0

t

r(t)

0 0.1 0.2 0.3 0.4 0.5

0

0.02

0.04

0.06

0.08

Sample paths of r(t) with γ = 0.5

t

r(t)

0 0.1 0.2 0.3 0.4 0.5

0

0.02

0.04

0.06

0.08

Sample paths of r(t) with γ = 1

t

r(t)

0 0.1 0.2 0.3 0.4 0.5

0

0.02

0.04

0.06

0.08

Sample paths of r(t) with γ = 1.5

tr(

t)

Figure 4: Bundles of simulated paths for the interest rate process (17)for various values of . The parameters used are calibrated to theestimates of Chan et al. (1992)

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−0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.120

10

20

30

40

50

60

70

80

r(0.5)

Fre

q

γ = 0γ = 0.5γ = 1γ = 1.5

Figure 5: The density functions for r at t = 0.05 corresponding to thesimulations in Fig.4

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Chan, Karolyi, Longstaff, and Sanders (1992) have estimateda discretised version of this model on US Treasury bill data forthe period 1964 to 1989 and found (in terms of our notation)� = 0.5921, r = 0.0689, �2 = 1.6704 and = 1.4999.

The value of �2 may seem high but it should be borne in mindthat the average volatility is measured by�r = 0.0234(= 2.34%p.a.) which is a reasonable value forinterest rate markets.

Noting the interpretation that 1/� is the average time forreversion back to the mean we see that the estimated value of� = 0.5921 implies that this average time is about 1.69 years.This value also looks reasonable.

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In Fig.4 we illustrate simulations of r given by (17) for varyingvalues of which use the same sequence of random numbers.

In these simulations we have used the values � = 0.6, r = 0.07(which are close to those found byChan, Karolyi, Longstaff, and Sanders (1992)) and the valuesof � displayed in Table 2.2.1.

It is of interest to note that when = 0, interest rates candrift very close to zero or even become negative though theprobability of this event is very low

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As ≥ 1/2 the bundle of interest rate paths moves away from0.

0 0.5 1 1.5

� 0.0234 0.08844 0.33429 1.26348

Table 2.1: Values of � and used in simulations of the interest rateprocess; here �r = 2.34%

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Hence a commonly used form of the volatility function is (16)with = 1/2 and this is the basis of the Cox-Ingersoll-Rossmodel.Stochastic processes having a linear drift term such as (15)and a square root volatility have been extensively investigatedby Feller (1951);∼ showed that the conditional transition density function for eqn.

(14) with

�(r, t) = �(r − r), �(r, t) = �√r, (18)

is given by

p(r, t∣r0, 0) = gt

∞∑

n=0

e−�t/2(�t/2)n(1/2)n+d/2

n!Γ(n+ d/2)(gtr)

n−1+d/2e−(gtr)/2

≡ gtP�2(d,�t)(gtr), (19)

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where P�2(d,�)(x) is the density function of a non-centralChi-squared distribution with d degrees of freedom andnon-centrality parameter �.Alternatively, the density function may be expressed as (which isreally the form given by Feller)

p(r, t∣r0, 0) = gte−(gtr+�t)(gtr)

(d−1)/2√�t

2(�tgtr)d/4Id/2−1(

√

�tgtr), (20)

where the modified Bessel function of the first find I�(x) is definedas

I�(x) =∞∑

n=0

(x/2)2n+�

n!Γ(n+ �+ 1).

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In the above formula the parameters d, gt and �t are defined as

d = 4�r/�2,

gt =4�

�2(1− e−�t),

�t =4�r0

�2(e�t − 1).

We need the condition 2�r > �2 to ensure that r is positive.

A plot of this p(r, t∣r0, 0) for a range of values � and thevalues of �, r indicated earlier is displayed in Fig.6.

This result is valid for 0 ≤ �r ≤ �.

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0 0.05 0.1 0.150

10

20

30

40

50

60σ = 0.06σ = 0.08844σ = 0.14σ = 0.20

Figure 6: The transition density function of the Feller process fordifferent values of �. The remaining parameters of the processcorrespond to those used for the simulations in Figure 4

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An important criticism of modelling the volatility by use offunctional forms such as (16) is that volatility of interest ratesdepend on the level of interest rates.

∼ This seems to run against what is often observed as it ispossible to identify historical periods when rates were high butrelatively stable or rates were low but fairly volatile.

∼ It is also often argued that volatility of interest rates shouldalso be a function of the news arrival process (i.e. the dW ’s).

∼ One way to capture this effect would be to allow � to itselffollow a stochastic process - in this regard see Problem 23.8.

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Motivating the Feller (or Square Root) Process


A very widely used stochastic process in finance is the Felleror square root process, frequently referred to as the Cox,Ingersoll and Ross (or just CIR) process.

Its interest for finance applications lies in the fact that (forcertain parameter constellations) the process is guaranteed togenerate positive values, and its density function is known.

This is clearly a desirable feature for a model of interest rateprocesses.

In this section we seek to give some insight into why thisprocess generates positive (or at least non-negativeoutcomes), and also why its conditional transitionalprobability density function is a chi-squared distribution.

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Consider the process (14) when �(r, t) = (�− �r) and�(r, t) = �(t), i.e.

dr = (�− �r)dt + �(t)dW (t). (21)

This is essentially an O-U process (Section 6.3.5) whose solution is

r(t) = r0e−�t +

∫ t

0

�e−�(t−s)ds+

∫ t

0

e−�(t−s)�(s)dW (s). (22)

It follows from (22) that

r(t) ∼ N(M(t), V 2(t)), (23)

∼ where

M(t) = r0e−�t +

∫ t

0

�e−�(t−s)ds,

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∼ and

V 2(t) = E0

[

(∫ t

0e−�(t−s)�(t)dW (s)

)2]

=

∫ t

0e−2�(t−s)�2(s)ds.

The fact that r(t) is normally distributed means that there isa positive probability that r can become negative asillustrated in Fig.7.

∼ This is not a desirable feature of a process for interest ratedynamics, though in practice this probability may be quite lowas can be seen from the simulated distribution in Fig.5 for thevalue = 0.

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r

Prob.

M(t)

Figure 7: The possibility of negative interest rates from the distributionin (23)

We are therefore motivated to find a process that has zeroprobability of r becoming zero or negative.

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Consider the set of u (mean reverting to zero) O-U processes

dXj = −1

2�Xjdt+

1

2�dWj , (j = 1, ⋅ ⋅ ⋅ , u). (24)

Integrating (24) we obtain

Xj(t) = Xj(0)e−

12�t +

1

2�

∫ t

0e−

�2(t−v)dWj(v), (25)

∼ implying that

Xj ∼ N(MXj(t), VX(t)),

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whereMXj

(t) = Xj(0)e−

12�t,

and

VX(t) =�2

4

∫ t

0e−�(t−v)dv.

Consider the process

r(t) = X1(t)2 +X2(t)

2 + ⋅ ⋅ ⋅+Xu(t)2.

Ito’s lemma ⇒

dr =

(

u�2

4− �r

)

dt+ �√r(

u∑

i=1

Xi√rdWi). (26)

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Consider the new process

W (t) =u∑

i=1

∫ t

0

Xi(v)√

r(v)dWi(v), (27)

∼ so that

dW (t) =

u∑

i=1

Xi(t)√

r(t)dWi(t).

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We calculate5

E[dW (t)] = 0,

∼ and

var[dW (t)] =

u∑

i=1

X2i (t)

r(t)E[dW 2

i ]

= (

u∑

i=1

X2i )

dt

r= dt.

Furthermore we easily calculate that

E[(dW (t))m] = 0 for m ≥ 3.

So dW (t) has all the statistical properties of a Wiener increment.

It thus follows that W (t) is a Wiener process.5Recall that E(dWidWj) = 0 (i ∕= j) (j = 1, 2, ⋅ ⋅ ⋅ , d).

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So we can write (26) as

dr =

(

u�2

4− �r

)

dt+ �√rdW, (28)

∼ which enjoys the property r ≥ 0 by construction.

The process (28) is simply the Feller process alreadyencountered in Section 22.2.

∼ It is more simply written here as

dr = (�− �r)dt + �√rdW, (29)

∼ where we choose u as

u =4�

�2> 0. (30)

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Now, for arbitrary � and �, u is not an integer.

∼ It can be shown that results for the CIR process with u equalto an integer are also true for u not an integer.

The value of u for the process affects very much its behaviour.We distinguish two cases:

u < 2 ⇒ � < 12�

2

In this case it can be shown that

Pr

{

r(t) = 0 at infinite numberof values of t

}

= 1.

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This implies behaviour shown in Fig.8.

∼ The excursions at r = 0 are also not a desirable feature of aprocess for interest rate dynamics since it does not accord withwhat we observed empirically.

t

r

Figure 8: The Feller process for r with � < �2/2

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u > 2 ⇒ � ≥ 12�

2

In this case

Pr {∃ at least one t > 0 s.t. r(t) = 0} = 0

∼ which is the desirable feature of the process for r that we seek.∼ Fig.9 illustrates what a typical sample path would look like.

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t

r

Figure 9: The Feller process for r with � > �2/2

In order to give an intuitive feel for the result that the distn.for r is a non-central Chi-squared distn., we set

X1(0) = X2(0) = ⋅ ⋅ ⋅Xu−1(0) = 0

Xu(0) =√

r(0).42 / 77




Then from (30) it follows that

Xi(t) ∼ N(0, V 2X(t)) i = 1, ⋅ ⋅ ⋅ , u− 1,

∼ and soXu(t) ∼ N(e−

12�t

√

r(0), V 2X(t)).

Thus

r(t) = V 2X(t)

u−1∑

i=1

(Xi(t)

VX(t))2 +X2

u(t).

Note that X2u(t) is the square of a normally distributed variable with

non-zero mean and that each of the (u− 1) terms in the sum on theRHS is the square of a (zero-mean) normally distributed variable;

∼ it follows that r(t) has a non-central Chi-squared distn. with

u− 1 = 4�−�2

�2 degrees of freedom.∼ The actual density function is derived at eqn. (22.20).

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Fubini’s Theorem

Fubini’s Theorem

We shall frequently encounter stochastic double integrals ofthe form

∫(∫

g(u, v, !)dW (u)

)

dv, (31)

with a variety of different limits in the integrals.

∼ Here ! can be any stochastic variable such as the forward rate,the bond price or the spot interest rate.

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Fubini’s Theorem

Fubini’s Theorem

We will very often need to calculate the stochastic differentialof terms like (31) and this is more simply done if we canexpress it in the form in the form

∫(∫

g(u, v, !)dv

)

dW (u). (32)

In other wordswe need to change the order of integration in stochasticdouble integrals.

The result that allows us to do this under fairly mild conditions on gis Fubini’s theorem.

Essentially this result allows us to manipulate stochastic doubleintegrals in the same way that we manipulate ordinary doubleintegrals.

Three main results may be derived from Fubini’s theorem, these are:

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Fubini’s Theorem

Fubini’s Theorem

(I)∫ �

0

(∫ t

0g(u, v, !)dW (u)

)

dv =

∫ t

0

(∫ �

0g(u, v, !)dv

)

dW (u),

(33)

(II)∫ T

t

(∫ �

0g(u, v, !)dW (u)

)

dv =

∫ �

0

(∫ T

tg(u, v, !)dv

)

dW (u).

(34)

In these two results the integration limits do not involve theintegration variables u or v.

These results state that in this case we may change the order ofintegration as we do for the same situation with ordinary integrals.

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Fubini’s Theorem

Fubini’s Theorem

(III)∫ t

0

(∫ v

0g(u, v, !)dW (u)

)

dv =

∫ t

0

(∫ t

ug(u, v, !)dv

)

dW (u).

(35)

Now the integration limits depend on the integration variables,

∼ the result again shows that we can handle such stochasticdouble integrals in the same way that we handle ordinarydouble integration.

∼ Figures 10, 11 and 12 illustrate the relevant regions ofintegration for cases I, II and III respectively. In each case theregion of integration (the shaded area) can be swept out byleft to right motion of the vertical line (the LHS doubleintegrals) or, from top to bottom motion of the horizontal line(the RHS double integrals).

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Fubini’s Theorem

Fubini’s Theorem

vu = 0 �

u

t

v = 0 v = �

u = t

Figure 10: Change of limits for version I of Fubini’s theorem

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Fubini’s Theorem

Fubini’s Theorem

vt u = 0 T

u

�

v = t v = T

u = �b

Figure 11: Change of limits for version II of Fubini’s theorem

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Fubini’s Theorem

Fubini’s Theorem

vu = 0 t

u

t

v = u

u = v

v = t

v = u

Figure 12: Change of limits for version III of Fubini’s theorem

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Modelling Forward Rates


Recall that f(t, T ) denotes the forward rate at time t forinstantaneous borrowing at time T .

Heath, Jarrow, and Morton (992a) propose to model theforward rate as the driving stochastic process.

They write the process for f(t, T ) in the form of a stochasticintegral eqn., i.e.,

f(t, T ) = f(0, T ) +

∫ t

0�(u, T, !(u))du +

∫ t

0�(u, T, !(u))dW (u),

(36)

for 0 ≤ t ≤ T .

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Ttt+ dt

f(t; T ) + �(t; T )dtdWf(t; T )f(t; T ) + �(t; T )dt+ �(t; T )dW

�(t; T )dW

1

Figure 13: Evolution of the forward rate curve

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The �(u, T, !(u)) and �(u, T, !(u)) are the drift andvolatility of the forward rate process.

f(0, T ) is the initial forward rate curve – can be obtainedfrom currently observed yields.

In a more general notation one would write �(u, T, !(u)) and�(u, T, !(u)). The ! indicates possible dependence on pathdependent quantities without specifying precisely what thedependence is.

We can allow for possible dependence of � and � on r(t) tocapture the effect of the level of interest rates on volatility.

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In the discussion of this section we only allow one shock termdW in order to alleviate the mathematical notation.

In differential form we may write eqn. (36) as the s.d.e.

df(t, T ) = �(t, T, !(t))dt + �(t, T, !(t))dW (t). (37)

We shall defer for the moment discussion of appropriatefunctional forms for �(t, T, !) and �(t, T, !).

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Fig.13 illustrates how the forward curve at time (t+ dt)evolves from that at t by the addition of two terms:

− The first, �(t, T, ⋅)dt, represents the average change across thematurities.

− The second, �(t, T, ⋅)dW , represents the impact at differentmaturities of the shock dW that occurs during (t, t+ dt).

The aim of the calculations in the rest of this section is todetermine the stochastic processes for the spot rate r(t) andbond price P (t, T ) implied by eqn. (36).

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The reader needs to be wary that the correct path from theforward rate dynamics to the spot rate dynamics has to occurat the level of the stochastic integral equations.

Thus we must set T = t in the stochastic integral equation(36) for f(t, T ) to obtain the stochastic integral equation forr(t).

From this latter we can obtain the stochastic differentialequation for r(t).

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It is not correct to obtain the stochastic differential equationfor r(t) simply by setting T = t in the stochastic differentialequation (37) for f(t, T ).

This has to do with the fact that the path history mattershere and we need to work with the stochastic integralequations, which are properly defined mathematical objects.

The stochastic differential equation is merely a (veryconvenient) short-hand notation as we stressed in Chapter 4.

We illustrate the correct path from forward rate dynamics tospot rate dynamics in Fig.14.

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df = �dt+ �dWNo

T = tdr = [⋅ ⋅ ⋅ ]dt+ �(t, t, ⋅)dW (t)

f(t, T ) = f(0, T ) +∫ t

0�(s, T, ⋅)ds+

∫ t

0�(s, T, ⋅)dW (s)

T = t

r(t) = f(0, t) +∫ t

0�(s, t, ⋅)ds+

∫ t

0�(s, t, ⋅)dW (s)

Figure 14: The path from forward rate dynamics to spot rate dynamics

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Recalling that the spot rate of interest and forward rate are relatedby r(t) = f(t, t) we see from eqn. (36) thatr(t) satisfies the stochastic integral eqn.

r(t) = f(0, t) +

∫ t

0

�(u, t, !(u))du+

∫ t

0

�(u, t, !(u))dW (u). (38)

This may alternatively be expressed as the s.d.e. (see section 5.5)

dr =

[

f2(0, t) + �(t, t, !(t)) +

∫ t

0

�2(u, t, !(u))du

+

∫ t

0

�2(u, t, !(u))dW (u)

]

dt+ �(t, t, !(t))dW (t).

(39)

Here �2, �2 denote partial differentiation of �, � with respect totheir second arguments.

59 / 77




Close inspection of (39) reveals that this process for r(t) ismore general than the types of processes considered in Section22.2.

∼ The difference stems from the third component of the driftterm, viz.,

∫ t

0

�2(u, t, !(u))dW (u). (40)

This term is a weighted sum of the shock terms dW (u) frominitial time 0 to current time t, thus,it is a path-dependent term.

∼ Hence the s.d.e. for r(t), eqn. (39) is non-Markovian6.6In fact even without the term (40) we have non-Markovian dynamics

because of the term∫ t

0�2(u, t, !(u))du, which depends on the path followed

by r between 0 and t.60 / 77



From Forward Rate to Bond Price Dynamics


In this section we determine the bond price dynamics impliedby the forward rate dynamics (36).

Before proceeding we compute a quantity that will be usefulin simplifying the expression for the bond price that we shallcalculate below. We note from eqn. (38) that

∫ t

0r(s)ds =

∫ t

0f(0, s)ds+

∫ t

0

∫ s

0�(u, s, !(u))duds

+

∫ t

0

∫ s

0�(u, s, !(u))dW (u)ds,

61 / 77





∼ which by use of result (III) of Fubini’s Theorem can be written∫ t

0

r(s)ds =

∫ t

0

f(0, s)ds+

∫ t

0

(∫ t

u

�(u, s, !(u))ds

)

du

+

∫ t

0

(∫ t

u

�(u, s, !(u))ds

)

dW (u).

(41)

In order we determine the s.d.e. for the bond price that isimplied by the forward rate process (36), we recall first thedefinitional relationship between bond prices and forwardrates, viz.,

P (t, T ) = exp

(

−∫ T

tf(t, s)ds

)

,

or

lnP (t, T ) = −∫ T

tf(t, s)ds. (42)

62 / 77





Substituting eqn. (36) into this last eqn. we have that

lnP (t, T ) =−∫ T

tf(0, s)ds−

∫ T

t

∫ t

0�(u, s, !(u))duds

−∫ T

t

∫ t

0�(u, s, !(u))dW (u)ds.

Applying result (II) of Fubini’s Theorem we can re-expresslnP (t, T ) as

lnP (t, T ) =−∫ T

tf(0, s)ds −

∫ t

0

(∫ T

t�(u, s, !(u))ds

)

du

−∫ t

0

(∫ T

t�(u, s, !(u))ds

)

dW (u),

63 / 77





which may further be re-arranged to7

lnP (t, T ) =−

∫ T

0

f(0, s)ds−

∫ t

0

(∫ T

u

�(u, s, !(u))ds

)

du

−

∫ t

0

(∫ T

u

�(u, s, !(u))ds

)

dW (u)

+

∫ t

0

f(0, s)ds+

∫ t

0

(∫ t

u

�(u, s, !(u))ds

)

du

+

∫ t

0

(∫ t

u

�(u, s, !(u))ds

)

dW (u). (43)

7

Since

∫

T

tf(0, s)ds =

∫

T

0f(0, s)ds −

∫

t

0f(0, s)ds, and for any function g we have

∫

t

0

(∫

T

tg(u, s)ds

)

du =

∫

t

0

(∫

u

tg(u, s)ds +

∫

T

ug(u, s)ds

)

du

=

∫

t

0

(

−

∫

t

ug(u, s)ds +

∫

T

ug(u, s)ds

)

du.

64 / 77





From eqn. (41) we see that we can represent the last three

terms of (43) as∫ t0 r(s)ds,

∼ from (42) we see that

−∫ T

0

f(0, s)ds = lnP (0, T ).

Hence equation (43) simplifies to

lnP (t, T ) = lnP (0, T ) +

∫ t

0r(s)ds−

∫ t

0

(∫ T

u�(u, s, !(u))ds

)

du

−∫ t

0

(∫ T

u�(u, s, !(u))ds

)

dW (u).

(44)

65 / 77





Taking differentials, the log bond price B(t, T ) ≡ lnP (t, T )satisfies the s.d.e.

dB(t, T ) = [r(t)− �B(t, T )] dt+ �B(t, T )dW (t), (45)

∼ where we define

�B(t, T ) =

∫ T

t

�(t, s, !(t))ds and �B(t, T ) = −∫ T

t

�(t, s, !(t))ds.

(46)

By Ito’s Lemma (since P (t, T ) = eB(t,T ))

dP (t, T ) =

[

r(t)− �B(t, T ) +1

2�2B(t, T )

]

P (t, T )dt

+ �B(t, T )P (t, T )dW (t). (47)

66 / 77





The s.d.e.s (39) and (47) form a linked system for theinstantaneous spot rate and the bond price.

This system will typically be non-Markovian and manyapplications in the HJM framwork seek simplifications whichreduce this system to Markovian form.

We give one example of how this may be done in the nextsection.

Since the steps from the forward price dynamics to the bondprice dynamics are rather round-about we feel it may be usefulto summarise these in Fig.15.

67 / 77





f(t, T ) = f(0, T ) +∫ t0 �(u, T, ⋅)du +

∫ t0 �(u, T, ⋅)dW (u)

r(t) = f(0, t) +∫ t0 �(u, t, ⋅)du +

∫ t0 �(u, t, ⋅)dW (u)

B(t, T ) = B(0, T ) +∫ t0 r(s)ds −

∫ t0 �B(s, T )ds −

∫ t0 �(s, T )dW (s)

dB(t, T ) = [r(t) − �B(t, T )] dt + �B(t, T )dW (t)

dP (t,T )P(t,T )

=(

r(t) − �B(t, T ) + 12�2B(t, T )

)

dt + �B(t, T )dW (t)

T = t

P (t, T ) = exp(

−

∫ T

tf(t, s, ⋅)ds

)

B(t, T ) = lnP (t, T ) = −

∫ Tt

f(t, s, ⋅)ds

P (t, T ) = eB(t,T )

Figure 15: The steps from forward rate to bond price dynamics 68 / 77





00.1

0.20.3

0.40.5

0

0.5

1

1.5

2

2.5

30.04

0.05

0.06

0.07

0.08

0.09

0.1

t (Years)

Simulated Evolution of f(t,T); 0 < t < 0.5; 0 < T < 3

T (Years)

f(t,T

)

0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.950

5

10

15

20

25

30Distribution of P(t,2) at t = 0.5

P(t,2)

f(P

(t,2

))

TheoreticalEmpirical

Figure 16: Simulations of the process for forward rates and bond prices

69 / 77



A Specific Example

A Specific Example

For the forward rate volatility in eqn. (37) assume thefunctional form

�(u, t, r) = �e−�(t−u)r(u) . (48)

Note that in this case

�2(u, t, r(u)) = −��e−�(t−u)r(u) = −��(u, t, r), (49)

70 / 77



A Specific Example

A Specific Example

so that the non-Markovian term in eqn. (39) may be written∫ t

0�2(u, t, r(u))dW (u) = −�

∫ t

0�(u, t, r(u))dW (u)

= −�

∫ t

0�e−�(t−u)r(u) dW (u).

(50)

Define the subsidiary stochastic variable

�(t) =

∫ t

0�e−�(t−u)r(u) dW (u), (51)

∼ and note that

d� = −��dt+ �r(t) dW (t). (52)

71 / 77



A Specific Example

A Specific Example

Using � to substitute the non-Markovian term in eqn. (39) wesee that r and � form a linked Markovian system viz.

dr = [f2(0, t) + �(t, t, r) +

∫ t

0�2(u, t, r)du − ��]dt+ �r(t) dW (t),

(53)

d� = −��dt+ �r(t) dW (t). (54)

In this system the past levels of volatility and the news arrivalprocess affect the dynamics of r via the variable �.

∼ This captures the type of process estimated byBrenner, Harjes, and Kroner (1996) though in a different wayto their specification.

∼ Here the drift rather than the volatility is affected directly bythe � process.

72 / 77



A Specific Example

A Specific Example

0 0.5 1 1.5 2 2.5 30.04

0.05

0.06

0.07

0.08

0.09

0.1

time t

r(t)

0 0.5 1 1.5 2 2.5 3−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

time tph

i(t)

Figure 17: Simulations of r(t) and �(t) for (53)–(54) with drift specifiedby (55)

73 / 77



A Specific Example

A Specific Example

In Fig.17 we illustrate some simulations of r(t) in the system(53)-(54) for = 0.5, 1.0 and 1.5. For the drift term we havetaken

�(t, T, r) = �(t, T, r)

∫ T

t�(t, u, r(t))du, (55)

∼ which ensures that the economy is arbitrage free as we shallshow in a later chapter.

∼ The values of �, � were taken from Bhar and Chiarella (997b).The initial forward rate curve f(0, t) is relevant to 90-day bankbill futures data traded on the SFE in 1991, and wascalculated using the polynomial fitting procedure described inBhar and Hunt (1993).

74 / 77



A Specific Example

Appendix 1. Calculation of Covariance

75 / 77



A Specific Example

Calculation of Covariance

Consider the interest rate process (28) in the case of adeterministic drift and volatility so that

r(t) = f(0, t) +

∫ t

0�(u, t)du +

∫ t

0�(u, t)dW (u).

We can readily calculate that

mr(t) = E[r(t)] = f(0, t) +

∫ t

0�(u, t)du.

We can then calculate the covariance between interest ratesat two maturities s and t as

cov(r(s), r(t)) = E0[{r(s)−mr(s)}{r(t)−mr(t)}]

= E0

[∫ s

0�(u, s)dW (u)

∫ t

0�(u, t)dW (u)

]

.

75 / 77



A Specific Example


For definiteness suppose s < t

E0

[∫ s

0�(u, s)dW (u)

∫ t

0�(u, t)dW (u)

]

= E0

[∫ s

0�(u, s)dW (u)

∫ s

0�(u, t)dW (u)

+

∫ s

0�(u, s)dW (u)

∫ t

s�(u, t)dW (u)

]

= E0

[∫ s

0�(u, s)dW (u)

∫ s

0�(u, t)dW (u)

]

=

∫ s

0�(u, t)2du.

76 / 77



A Specific Example


In general (i.e. s < t or s > t) we have

cov(r(s), r(t)) =

∫ min(s,t)

0�(u, t)2du

=

∫ s∧t

0�(u, t)2du

where s ∧ t ≡ min(s, t).

77 / 77


References

Bhar, R. and C. Chiarella (1997b).Interest Rate Futures: Estimation of Volatility Parameters in an

Arbitrage-Free Framework.Applied Mathematical Finance 4(4), 181–199.

Bhar, R. and B. Hunt (1993).Predicting the Short-Term Forward Interest Rate Structure

Using a Parsimonious Model.Review of Futures Markets 12, 577–590.

Brenner, R. J., P. H. Harjes, and K. F. Kroner (1996).Another Look of Models of the Short-Term Interest Rate.Journal of Financial and Quantitative Analysis 31, 85–107.

Chan, K. C., G. A. Karolyi, F. A. Longstaff, and A. B. Sanders(1992).

An Empirical Comparison of Alternative Models of theShort-Term Interest Rate.

77 / 77



A Specific Example

Journal of Finance 47, 1209–1227.

Feller, W. (1951).Two Singular Diffusion Processes.Annals of Mathematics 54, 173–182.

Heath, D., R. Jarrow, and A. Morton (1992a).Bond Pricing and the Term Structure of Interest Rates: A New

Methodology for Continent Claim Valuations.Econometrica 60(1), 77–105.

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